Nonlinear-optical susceptibility of hilgardite-like borates M2B5O9X (M=Pb,Ca,Sr,Ba ; X=Cl,Br )

Solid State Sciences 7 (2005) 1194–1200 www.elsevier.com/locate/ssscie

Nonlinear-optical susceptibility of hilgardite-like borates M2B5O9 X (M = Pb, Ca, Sr, Ba; X = Cl, Br) Paul A. Plachinda a,∗ , Valery A. Dolgikh b , Sergey Yu. Stefanovich b , Petr S. Berdonosov b a Department of Material Sciences, MSU, Russia b Department of Chemistry, MSU, Russia

Received 3 November 2004; received in revised form 17 May 2005; accepted 26 May 2005 Available online 28 July 2005

Abstract Usage of non-centrosymmetric boron oxide derivative crystals as nonlinear elements may be a key solution of pressing problem of obtaining powerful radiation at shorter wavelengths via frequency multiplication of solid state laser radiations. It is shown that Pb2 B5 O9 Br and some other members of hilgardite-like borohalogenides demonstrate in powder form, high non-linear optical activity in the SHG experiments. An attempt is made using the Philips–Van Vechten–Levine–Xue bond theory to calculate second-order nonlinear coefficients for isostructural to mineral hilhardite noncentrosymmetric crystals M2 B5 O9 X with M = Pb, Sr, Ba, Eu, Ca and X = Cl, Br. Theoretically obtained values of nonlinear optical coefficients dij k only partly correlate with the results of SHG measurements. In particular, the experiments distinctively show sharp increase of nonlinear activity along the series Ca < Sr < Ba < Pb and Cl < Br of metal boro-halogenides, while theoretical estimations give substantially more slack composition dependences. Analysis of contribution of different boron–oxygen bonds to optical nonlinearities of the compounds reveals important role of planar BO3 triangles, though this factor is yet unable to explain enormously high SHG output from Pb2 B5 O9 Br.  2005 Elsevier SAS. All rights reserved. Keywords: Borates; Non-linear optical susceptibility; Calculation; Measurements

1. Introduction Generation of second optical harmonic historically appeared to be the first effect among nonlinear optical (NLO) phenomenon, discovered in studying interaction of laser radiation with media. This effect instantly drew material scientists’ attention, as it was opening the possibility to create devices that would broaden the range of “working” laser frequencies at use of “standard” sources of radiation. Currently this principle is seemed unique for creation of solid state laser generators in the UV range of spectrum where direct laser emission in solid-state medium is difficult. Nowadays borate crystals, characterized by the recordbreaking wide window of a transparency, high share (36%) * Corresponding author. Tel.: +7 095 939-35-04.

E-mail address: [email protected] (P.A. Plachinda). 1293-2558/$ – see front matter  2005 Elsevier SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2005.05.006

of nonsymmetrical structures and high beam stability [1], are considered the key material for high-power UV generation of coherent radiation. However, the values of NLO susceptibility of borate crystals are usually significantly lower than the ones of the best NLO materials: LiNbO3 , KTiOPO4 , and others. In this connection, it is especially interesting to investigate origin of recently discovered abnormally high intensity of optical signal at doubled frequency, generated by powder of Pb2 B5 O9 Br [2] under Nd:YAG-laser irradiation (λ = 1064 nm). The lead bromine pentaborate, Pb2 B5 O9 Br, possesses the hilgardite type structure (Fig. 1(a)) [2], which is based on crossing chains of top-connected, BO4 and BO3 polyhedrons (Fig. 1(b)) [2]. The chains form a framework with go-through channels where lead and bromine atoms are located (Fig. 1(a)). Anhydrous hilgardite phases M2 B5 O9 X are currently known for M = Pb, Sr, Ca, Ba, Eu; X = Cl, Br, OH [2–6],

P.A. Plachinda et al. / Solid State Sciences 7 (2005) 1194–1200

(a)

1195

by tetrahedral BO4 and triangle BO3 groups [17]. In the quoted article, authors adduce values of dij k for Ca2 B5 O9 Cl and Eu2 B5 O9 Br with hilgardite-type structure. But they do not discuss the contribution of every bond into integral NLO susceptibility. As far as we know, the investigation of experimental SHG ability of hilgardite-type compounds was not carried out before. In this article, we make an attempt to discuss interrelation between chemical compositions of hilgardite-like compounds M2 B5 O9 X with M = Sr, Pb, Ba, Ca; X = Cl, Br and their NLO activities measured under standardized conditions, on one hand, and the corresponding theoretical χ values calculated from the PVL model, on the other hand. The choice of the objects for study was determined by expediency of experimental and theoretical comparison of NLO characteristics of hilgardite-like compounds. These substances include in their structure cations of various chemical nature and different halogens, considering presence of the complete structural data for reviewed phases.

2. Experimental 2.1. Synthesis

(b) Fig. 1. Crystal structure of lead bromine pentaborate [2]. (a) Polyhedral representation of framework formed by plane BO3 triangles and tetrahedron BO4 groups and extraframework atoms of Pb and Br. (b) Chemical bonding in B5 O9 radical.

but crystal structure details are available only for those containing the M–X pairs: Pb–Br [2], Ca–Br [7], Ba–Cl [8], Eu–Br, Eu–Cl [9]. Ab initio calculations of components of NLO susceptibility tensor (dij k ) for hypothetical structures are not yet possible. Theoretical interpretation of NLO characteristics of simpler borates is usually based on the anion group theory [10]. However, this approach cannot take into account many details of the structure of complex borates comprising more than one type of anions. From this point of view the Philips– Van Vechten–Levin’s model (PVL) [11–14] seems to be the most appropriate tool for these calculations. Xue et al. [15–17] successfully applied it for a variety of complex crystals with different structures. Good correlations of theoretical estimations with empirical dij k values are seen in those cases when calculation relies on precise structural data. Especially it was on example borates, including isolated planar BO3 groups, or different types of complex anions, formed

Initial reactants B2 O3 , H3 BO3 , SrO, PbO, PbX2 (X = Cl, Br), MCO3 (M = Ca, Ba) were used in solid state reactions, all of the grade “pure”. The Alkaline Earth halogenides were prepared by dehydration of corresponding crystalline hydrates at temperature 300 ◦ C under conditions of dynamic vacuum (10−1 –10−3 Torr). Because of strong hygroscopic properties of B2 O3 , we replaced this simple oxide by substantially more stable double borates, SrB4 O7 or PbB4 O7 . These two borates were synthesized at 400 ◦ C (56 h) from stoichiometric mixtures of B2 O3 and PbO (yellow modification), or SrO. All powders were prepared in dry chamber. Using the reaction equations: MO + 2MX2 + 5MB4 O7 = 4M2 B5 O9 X (M = Sr, Pb; X = Cl, Br) or 3MCO3 + MX2 + 10H3 BO3 = 2M2 B5 O9 X + 15H2 O + 3CO2

(M = Ca, Ba)

we prepared initial mixtures of powders, which were then heat-treated according to the synthesis conditions given in Table 1. 2.2. Powder X-ray diffraction (PXRD) analysis All intermediate and final substances were identified by PXRD phase analysis using a Guinier camera (FR-552, Enraf-Nonius, CuKα 1-radiation, germanium as the internal standard) or Stoe STADI-P diffractometer. The Guinier

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Table 1 Conditions of synthesis of the hilgardite-like phases M2 B5 O9 Br (M = Sr, Pb, Ba; X = Cl, Br) No.

Initial mixture / compound

Container

Conditions of synthesis

1

2PbBr2 *PbO*5PbB4 O7 Pb2 B5 O9 Br

Alumna crucible in evacuated quartz ampoule

2

2PbBr2 *PbO*5PbB4 O7 Pb2 B5 O9 Br

Evacuated quartz ampoule

Annealing at 450 ◦ C (2 h), heating (100 ◦ C/h) till 650 ◦ C, annealing (24 h), cooling (3 ◦ C/h) till 400 ◦ C, cooling (switched-off furnace mode). Milling, annealing at 750 ◦ C (24 h). Annealing at 760 ◦ C (4 h), cooling (3 ◦ C/h) till 400 ◦ C, cooling (switched-off furnace mode).

3

2PbCl2 *PbO*5PbB4 O7

Evacuated quartz ampoule

Annealing at 450 ◦ C (2 h), heating (100 ◦ C/h) till 650 ◦ C annealing (24 h), cooling (switched-off furnace mode). Milling, annealing: 750 ◦ C (24 h). Annealing at 760 ◦ C (4 h), cooling (3 ◦ C/h) till 400 ◦ C, cooling (switched-off furnace mode).

Pb2 B5 O9 Br 4

2PbCl2 *PbO*5PbB4 O7 Pb2 B5 O9 Cl

Alumna crucible in evacuated quartz ampoule

5

2SrBr2 *SrO*5SrB4 O7 Pb2 B5 O9 Cl

Evacuated quartz ampoule

6

2SrCl2 *SrO*5SrB4 O7

Evacuated quartz ampoule

Annealing at 450 ◦ C (2 h), heating (100 ◦ C/h) till 650 ◦ C, annealing (24 h), cooling (3 grad/h) till 400 ◦ C, cooling (switched-off furnace mode). Milling, annealing: 750 ◦ C (24 h). Annealing at 450 ◦ C (2 h), heating (100 ◦ C/h) till 650 ◦ C, annealing (24 h), cooling (3 ◦ C/h) till 400 ◦ C, cooling (switched-off furnace mode). Milling, annealing: 750 ◦ C (24 h). Annealing at 300 ◦ C (3 h), heating (100 grad/h) till 1000 ◦ C annealing (2 h), cooling with rate 2 ◦ C/h till 700 ◦ C.

Sr2 B5 O9 Br 7 8 9 10

3BaCO3 *1BaBr2 *10H3 BO3 Ba2 B5 O9 Br 3BaCO3 *1BaCl2 *10H3 BO3

Flow of Ar

Ba2 B5 O9 Cl 3CaCO3 *1CaBr2 *10H3 BO3 Ca2 B5 O9 Br 3CaCO3 *1CaCl2 *10H3 BO3 Ca2 B5 O9 Cl

Flow of Ar

Annealing at 300 ◦ C (3 h), heating (100 grad/h) till 1000 ◦ C annealing (2 h), cooling with rate 2 ◦ C/h till 700 ◦ C.

Flow of Ar

Annealing at 300 ◦ C (3 h), heating (100 ◦ C/h) till 1000 ◦ C annealing (2 h), cooling with rate 2 ◦ C/h till 700 ◦ C.

Flow of Ar

Annealing at 300 ◦ C (3 h), heating (100 ◦ C/h) till 1000 ◦ C annealing (2 h), cooling with rate 2 ◦ C/h till 700 ◦ C.

Table 2 Unit cell parameters (S.g. Pnn2) for M2 B5 O9 X (M = Ca, Sr, Ba, Pb; X = Cl, Br) powders Compound Ca2 B5 O9 Cl Ca2 B5 O9 Br Sr2 B5 O9 Cl Sr2 B5 O9 Br Ba2 B5 O9 Cl Pb2 B5 O9 Cl Pb2 B5 O9 Br

M

Unit cell parameters, Å a

b

11.257(8) 11.39(7) 11.401(6) 11.5201(2) 11.635(8) 11.414(9) 11.510(1)

11.138(7) 11.25(3) 11.323(4) 11.4196(5) 11.581(7) 11.410(3) 11.419(1)

Table 3 SHG intensities for powdered M2 B5 O9 X hilgardites (α-SiO2 units) Ca

Sr

Ba

Pb

X c

2ω (L = 3–5 µm) I 2ω /ISiO

4

7

8

20

2ω (L = 3–5 µm) Br I 2ω /ISiO

5

12

32

80

Cl 6.12(9) 6.29(3) 6.504(6) 6.4868(4) 6.680(3) 6.583(6) 6.530(4)

X-ray diagrams were analyzed with use of home-made program, while diffractometer data were refined with help of the WinXpow program package [18]. In phase identification there was ICDD PDF2 database [19] employed. According to PXRD, experiments mentioned in Table 1 resulted in single-phase samples M2 B5 O9 X (M = Ca, Sr, Ba, Pb; X = Cl, Br). Their unit cell parameters (Table 2) appeared to be in agreement with those given in literature [19].

2

2

and T.T. Perry [20]. Graded powder samples were separated into fractions on sieves, according to particle size in samples. YAG:Nd-laser was used as source of powerful impulse radiation at wavelength λ = 1.064 µm with a repetition rate of 4 impulses per second and a duration of impulses about 10 ns. The intensities I2ω of the signals at doubled frequency (λ2ω = 0.532 µm) were registered from substances in the backward direction. Measured values are presented in Table 3 in relation to SHG intensity of the α-quartz 3-µm powder taken as standard.

3. Theoretical approach 2.3. Optical second harmonic generation The NLO properties of the M2 B5 O9 X compounds were studied by powder second harmonic generation (SHG) technique with the set-up similar to that employed by S.K. Kurtz

Strict approach to calculations of second order susceptibility tensor components should be based on knowledge of electronic band structure for a crystal. It is quite difficult to follow this way in the case of compounds with complicated

P.A. Plachinda et al. / Solid State Sciences 7 (2005) 1194–1200

structures. That is why so far the use of some simplified models is unavoidable. From a number of these models the Phillips–Van Vechten–Levine [11–14] is the most promoted. This electrodynamic model ascribes nonlinear terms in susceptibility to the anharmonic motion of bond charges q located approximately halfway between the two neighboring atoms. The description of the crystal is given by the average gap Eg between valence and conduction bands that can be separated into homopolar and heteropolar (i.e., covalent and ionic) contributions E h and C, such as Eg2 = Eh2 + C 2 .

(1)

For any bond between atoms α and β the equation for C is the Cαβ = 1/4πε0 be2 (Zα /rα − Zβ /rβ )e−ks R ,

µ

type µ per cm3 , βzzz is hyperpolarizability of bond type µ, µ µ µ µ 1 Gij k = nη λ αi (λ)αj (λ)αk (λ) presents directing cosines b

interrelations between µ- and λ-bonds. The complete expression for the total nonlinear susceptibility   Zαµ + Zβµ  µ dij k = Fµ µ µ fi Zα − Zβ µ 2   µ r0 µ µ + s(2s − 1) µ fc ρ µ r0 − rc µ

×

(5)

d µ —bond length of the µ type bonds; q µ —bond charge of the µth bond [7]; s—bond force constant (s = 2.48); µ µ µ rc = 0.35r0 —core radius, where r0 = d µ /2; µ µ µ µ ρ = (rα − rβ )/(rα + rβ )—difference in the atomic sizes.

(3)

where r is half of the α–β bond length and s = 2.48. The µ µ ionicity fi and covalency fc of the individual µ bond are given by µ

µ

fi = (C µ )2 /(Eg )2 ,

fcµ = (Eh )2 /(Eg )2 .

The linear susceptibility χ µ contributed for the bond type µ in the crystal is
2 χ µ = (4π)−1 hΩpµ /Egµ ,

(4)

µ

where Ωp is the plasma frequency (hΩp)2 = (N e2 /m)DA, N = nv /vb , n—the number of valence electrons on the bond between α and β, v—number of bonds type µ in unit cell, D, A—correction factors of order unity. The total susceptibility of the crystal composed of different µ types bonds is  µ µ  µ µ χ = F χ = Nb χb , where F µ —fraction of bonds of type µ composing the crystal; χ µ —linear susceptibility contribution from µ type bonds; µ Nb —number of bonds of type µ per cm3 ; µ χb —susceptibility of single bond of type µ. The second-order nonlinear optical tensor coefficients dij k could be written as following   µ µ µ µ dij k = F µ dij k = Gij k Nb βzzz , µ µ

ε0 (χb )2 µ µ G N , d µ q µ ij k b

where:

(2)

where R = 12 (rα + rβ ), rα , rβ —the covalent radius of atoms α and β respectively; b—dimensionless constant of order of unity; Zα , Zβ —effective number of valence electrons of A and B ions, respectively; (e)—electron charge. Eh ∝ r −s ,

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where dij k is the total macroscopic nonlinearity of a crysµ tal consists only of bonds type µ, Nb —number of bonds

In order to make the Levine model better applicable to complex crystals, Xue and Zhang [15,16] developed a method for decomposing of complex crystal lattice into systems of bonds between pairs of atoms. According to them, a crystal may be described by its ‘crystal formula’ as combination of numerous chemical bonds; different bonds have appropriate locations and lengths. Each type of bond has also its own element ratio and effective valence. The element ratio of a chemical bond is labeled by a formula, called bond subformula. The subformula for the bond A– B in a multi-bond crystal Aa Bb Cc Dd Gg . . . is expressed as follows:     N (A − B)b N (B − A)a A B, (6) NCA NCB where A, B, C, D, G, . . . represent different elements or different locations of the same element, N (B–A) represents the number of B-ions in first coordination sphere of an ion A, NCA is being coordination number of the ion A. Formula (6) can also be written as N (B − A)a AB N(A−B)bNCA , NCA N(B−A)aNCB where n=

N (A − B)bNCA N (B − A)aNCB

(6 )

is the ratio between elements A and B. Final result for dij k of a complex crystal could be rewritten as:   Zαµ + nZβµ  µ Fµ dij k = µ µ fi Zα − nZβ µ

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P.A. Plachinda et al. / Solid State Sciences 7 (2005) 1194–1200

 + s(2s − 1)

2

µ

µ

r0

µ

r0 − rc



+ 2/9Pb(1)O(5)9/3 + 2/9Pb(1)O(6)9/4

fcµ ρ µ

+ 2/9Pb(1)O(7)9/4 + 2/9Pb(1)O(9)9/4

µ

ε0 (χ )2 µ µ × µ bµ Gij k Nb . d q

+ 2/9Pb(1)Br(1)9/2 + 2/9Pb(1)Br(2)9/2

(5 )

+ 2/9Pb(2)O(1)9/4 + 2/9Pb(2)O(1)9/4 + 2/9Pb(2)O(1)9/4 + 2/9Pb(2)O(3)9/4

4. Second-order susceptibility of Pb2 B5 O9 Br

+ 2/9Pb(2)O(4)9/3 + 2/9Pb(2)O(6)9/4

The Pb2 B5 O9 Br crystal data is given in Table 2. According to Belokoneva et al. [2], there are 2 nonequivalent ions of lead, 5 borons, 9 oxygens, and 2 bromines in the unit cell, all bonds between them being non-equivalent. The same is true for all other members of the hilgardite structural family. In accordance with this, we can write the subformula for hilgardite-type compounds (using lead bromine pentaborate as an example) as following:

+ 2/9Pb(2)O(7)9/4 + 2/9Pb(2)O(8)9/3

Pb2 B5 O9 Br = 2/9Pb(1)O(1)9/4

+ B(4)O(4) + B(4)O(4) + B(4)O(8)3/4

+ 2/9Pb(2)Br(1)9/2 + 2/9Pb(2)Br(2)9/2 + B(1)O(1) + B(1)O(5)4/3 + B(1)O(6) + B(1)O(7) + B(2)O(3) + B(2)O(2)4/3 + B(2)O(7) + B(2)O(4)4/3 + B(3)O(1) + B(3)O(3) + B(3)O(9) + B(3)O(8)4/3 + B(5)O(2) + B(5)O(5) + B(5)O(9)3/4 .

+ 2/9Pb(1)O(2)9/3 + 2/9Pb(1)O(3)9/4

Table 4 The bond parameters and nonlinear properties of each type of bond in the Pb2 B5 O9 Br crystal and their contributions to the total nonlinear optical tensor coefficients µ

µ

µ

N

Bond

dµ, Å

Eh , eV

C µ , eV

fc

χµ

qµ, e

G311

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Pb1–O1 O2 03 05 06 O7 09 Br1 Br2 Pb2–O1 O3 04 O6 O7 O8 O9 Br1 Br2 B1–O1 O5 O6 O7 B2–O2 O7 O3 O4 B3–O1 O3 O8 O9 B4–O4 O6 O8 B5–O2 O5 O9

2.6455 2.5105 2.6774 2.5729 3.122 2.8038 2.9467 3.1228 2.9189 2.7393 2.9987 2.8465 2.6477 2.5531 2.6145 2.8349 3.0827 3.0707 1.4138 1.5312 1.492 1.4873 1.4902 1.5208 1.4489 1.4057 1.4639 1.5059 1.4127 1.4581 1.3751 1.3022 1.4211 1.3533 1.3201 1.3921

3.5596 4.0534 3.4552 3.814 2.3606 3.0819 2.7245 2.3593 2.7892 3.2649 2.6088 2.9684 3.5524 3.8876 3.6652 2.9987 2.4359 2.4597 16.8383 13.8155 14.7327 14.8486 14.7778 14.0494 15.8442 17.078 15.4434 14.3981 16.8684 15.596 18.0355 20.6456 16.622 18.7648 19.9592 17.4954

35.9534 33.2945 34.9544 31.4082 24.1513 31.3369 27.8106 45.5022 53.7591 33.1185 26.6544 24.5941 35.8851 39.0625 30.2267 30.522 46.9873 47.4468 17.2593 16.6866 15.4614 15.5621 17.6652 14.863 16.4188 19.9392 16.0758 15.1693 19.7355 16.2067 9.2938 9.6275 8.681 9.6049 10.1072 8.4122

9.71E−03 0.0146 9.68E−03 0.0145 9.46E−03 9.58E−03 9.51E−03 2.68E−03 2.68E−03 9.62E−03 9.49E−03 0.0144 9.70E−03 9.81E−03 0.0145 9.56E−03 2.68E−03 2.68E−03 0.4877 0.4067 0.4759 0.4765 0.4117 0.4719 0.4822 0.4232 0.4799 0.4739 0.4221 0.4808 0.7902 0.8214 0.7857 0.7924 0.7959 0.8122

0.0946 0.2177 0.0969 0.2286 0.1354 0.1066 0.1187 0.0287 0.0247 0.1016 0.1235 0.2829 0.0948 0.0882 0.2362 0.1092 0.0278 0.0276 3.0341 3.6542 3.2965 3.2804 3.5045 3.3961 3.1506 3.2072 3.2011 3.3443 3.2313 3.1816 6.4987 4.9992 6.9001 6.3125 6.0335 5.6491

1.5722 1.8227 1.5689 1.8064 1.5157 1.5551 1.5383 1.9169 1.9244 1.5623 1.5318 1.7299 1.572 1.5815 1.7953 1.5515 1.9185 1.9189 0.4799 0.5115 0.4477 0.4496 0.5302 0.4365 0.4652 0.5711 0.459 0.4423 0.5676 0.4613 0.232 0.2711 0.2126 0.2417 0.2572 0.232

−2.50E−04 0.0162 −2.68E−03 −0.0393 0.0731 0.0158 7.52E−03 0.2074 −0.252 0.1479 −0.1841 0.0397 0.2944 −0.3416 −0.1982 0.1563 −5.35E−03 8.70E−04 0.0506 −1.55E−03 −0.1402 −0.2886 −2.41E−03 5.07E−04 −0.2104 −0.3009 −0.0731 0.0231 −0.0917 −0.0848 −0.1875 −0.2976 0.1119 −0.0878 0.0866 −1.85E−03

All notations are described in Section 3.

µ

µ

δ311 , pm/V G322 −2.25E−09 −0.2112 6.17E−07 0.2353 −1.00E−07 −0.3793 −1.63E−06 −0.1356 4.73E−06 0.2216 6.87E−07 0.1389 3.91E−07 0.0776 4.29E−07 5.42E−03 −4.11E−07 −7.82E−03 5.96E−06 0.0211 −2.55E−06 −4.63E−03 2.38E−06 0.0184 1.06E−05 1.33E−06 −1.10E−05 −0.0203 −8.68E−06 −0.0458 7.08E−06 0.0483 −1.05E−08 −0.3664 1.69E−09 0.0774 3.71E−03 4.06E−03 −1.27E−04 −0.2132 −0.0142 −0.0376 −0.0288 −0.0963 −1.69E−04 −0.0804 1.44E−05 0.0374 −0.0179 −0.1355 −0.0147 −0.0224 −1.66E−03 −0.2996 2.47E−03 4.60E−03 −4.64E−03 −9.70E−05 −1.88E−03 −0.2531 0.1003 −1.21E−04 0.3497 −0.0528 −0.2794 0.0288 0.1745 −0.259 −0.0385 0.116 2.90E−03 −0.1794

µ

µ

δ322 , pm/V G333 −1.90E−06 −0.011 8.97E−06 0.5847 −1.42E−05 −0.236 −5.62E−06 −0.7224 1.44E−05 0.0363 6.05E−06 3.99E−03 4.04E−06 0.8693 1.12E−08 0.0113 −1.28E−08 −0.0225 8.48E−07 5.29E−03 −6.43E−08 −7.56E−03 1.10E−06 0.9114 4.79E−11 0.4985 −6.51E−07 −0.3272 −2.01E−06 −0.599 2.19E−06 9.88E−03 −7.23E−07 −0.1162 1.51E−07 4.89E−04 2.98E−04 0.9168 −0.0175 −0.0116 −3.82E−03 −6.24E−03 −9.60E−03 −0.1956 −5.64E−03 −5.80E−04 1.06E−03 0.9425 −0.0116 −0.074 −1.10E−03 −0.0535 −6.80E−03 −0.1187 4.93E−04 0.9582 −4.90E−06 −7.94E−04 −5.61E−03 −0.0658 6.49E−05 −7.41E−03 0.062 −0.3627 −0.072 0.7797 0.515 −0.3733 −0.0515 0.6743 0.281 −6.64E−03

µ

δ333 , pm/V −9.89E−08 2.23E−05 −8.81E−06 −2.99E−05 2.35E−06 1.74E−07 4.52E−05 2.33E−08 −3.67E−08 2.13E−07 −1.05E−07 5.47E−05 1.80E−05 −1.05E−05 −2.63E−05 4.47E−07 −2.29E−07 9.52E−10 0.0672 −9.53E−04 −6.34E−04 −0.0195 −4.07E−05 0.0267 −6.31E−03 −2.62E−03 −2.69E−03 0.1027 −4.01E−05 −1.46E−03 3.97E−03 0.4261 −1.9468 0.7424 −0.2995 0.0104

P.A. Plachinda et al. / Solid State Sciences 7 (2005) 1194–1200

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Table 5 Computed values dij k for hilgardite phases Compound

d31 (pm/V)

d32 (pm/V)

d33 (pm/V)

Eu2 B5 O9 Cl Eu2 B5 O9 Br Ca2 B5 O9 Cl Ca2 B5 O9 Br Sr2 B5 O9 Cl Sr2 B5 O9 Br Ba2 B5 O9 Cl Pb2 B5 O9 Br Pb2 B5 O9 Cl

0.2794 0.3004 0.1819 0.1956 0.2140 0.2261 0.2800 0.2389 0.2179

0.5302 0.5662 0.5183 0.5567 0.6215 0.6503 0.8447 0.6917 0.6418

−0.6559 −0.6535 −0.6616 −0.6907 −0.8332 −0.8589 −1.2025 −0.9310 −0.8825

Other variables necessary for the calculation algorithm were computed with the above-discussed formulas. The values used for dij k calculations for Pb2 B5 O9 Br are summarized in µ Table 4, where q µ is given in e (elemental charge) and δ3xx in pm/V. The results of computing of second-order susceptibility tensor components for Pb2 B5 O9 Br are presented in Table 5. To complete the picture, we carried out on the same methodology detailed calculations for M2 B5 O9 X (M = Sr, Ca, Ba, Eu; X = Cl, Br), having the sets of atomic positions found in literature. In case of unsolved structures (both for Sr and Ba), we assumed that the crystalline structure is the same as in Ca but the lattice parameters were taken from our experimental results. The Eu2 B5 O9 Br compound theoretically treated previously [17], and final value d333 we received for this bromine-containing borate is nearly the same as it had been obtained in [17]. Table 5 summarizes the results of the calculations.

5. Discussion It is seen from Table 5 that second-order optical nonlinearities for all hilhardite-like compounds do not significantly differ and are mainly about of the same order that crystalline quartz (d11 = 0.364 pm/V). The biggest nonlinearity is characteristic for Ba2 B5 O9 Cl and slightly lower for Pb2 B5 O9 Br, with very small decline in the case of its chlorine counterpart. Nevertheless, interrelations between theoretical values of the nonlinearity for different members of the family are not confirmed by the measurements of NLO activities of corresponding powders (Table 3). All compounds, mentioned above, are phase-matching. The measurements were carried out on powders of different particle size, and intensity of SH signal increased symbate to particle size. When comparing theoretical and experimental data on nonlinearity, it is necessary to take into account that SHG output in fine powders simply relates to intensity of light at doubled frequency through the relation:  3 n+1 I2ω /I2ω (SiO2 ), d = Ad111 (7) nSiO2 + 1 where A—geometrical factor close to unity, d111 = 0.364 pm/V; n, nSiO2 —refractive indices for the substances

(a)

(b)

(c) Fig. 2. Contribution of various bonds N (numeration corresponds to sequence of lines in Table 4) in nonlinear coefficients of Pb2 B5 O9 Br.

under investigation and quartz, while d stands for directionally averaged nonlinear coefficients, which can be roughly considered as mean quantity of all dij k values [21].

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P.A. Plachinda et al. / Solid State Sciences 7 (2005) 1194–1200

The main meaning factor in (7) is square root from relative SHG intensity the latter just being listed in Table 3. It is obvious that Table 3 figures only partly correlate with the results of the theoretical calculations of nonlinearities in M2 B5 O9 Cl and M2 B5 O9 Br series of hilgardite-like substances. dij k values computed by us for Eu2 B5 O9 Br and Ca2 B5 O9 Cl differ from values obtained in [17]. As it was already mentioned above, in this article only dij k values are listed without any description of the bonds (as they are listed in Table 4); therefore it is impossible to discuss the causes of divergence In particular, the experiments distinctively show monotonic rise of nonlinear activity of the powders along the series of metal halogen pentaborates in consequence: Ca < Sr < Ba < Pb. More than that, for halogen pentaborates of one and the same metal systematically exceeding of I2ω values for the bromides in comparison with the chlorides is noticed. This facts lead to a thought about very essential roles of extraframework cations and anions on NLO properties of these phases. For better understanding of origins and mechanism of influence of extraframework cations and anions on dij k , we carried out theoretical evaluation of the contribution of each bond (numbered in accordance with its position in Table 4) to the magnitudes d311 , d322 and d333 as it is shown by graphs in Figs. 2 (a), (b), (c). (On example of lead boron pentaborate.) Because of the strongest influence on optical nonlinearities of hilgardites, the bonds #32–35 are of the most interest. They correspond to boron–oxygen liaisons B(4)–O and B(5)–O, inside boron–oxygen triangles. Therefore, one could think that in hilgardites he finds one more confirmation of basic role of flat coordinated boron atoms inside BO3 polyhedrons in determining the NLO properties, as it is postulated in Chen’s theory [10]. Quite natural, however, that ordered arrangement of BO3 triangle plates along the polar axis is necessary to provide the borates high NLO susceptibilities [22]. This theoretical premise is being supported by the set of available experimental data. Keeping this in mind, it becomes harder to interpret the properties of hilgardites from the above position, because in these crystals the boron–oxygen chains tied with the tetrahedrons BO4 along polar c-axis, while the plates of triangles B(4)O3 and B(5)O3 are almost perpendicular to it and each other. The latter circumstance results in opposite signs of contributions of bonds B(4)–O and B(5)–O into integral magnitude d333 of the crystal. Unsatisfactory correspondence of experimental correlations between second-order optical activities of hilgarditefamily members with theoretically predicted values of

optical nonlinearity does not evidence against Philips– Van Vehthen–Levin–Xue approach to calculation of complex crystals nonlinearities. Sooner, more detailed electron density data are currently needed to supplement the picture of chemical bonds in such high nonlinear borates as Pb2 B5 O9 Br or BiB3 O6 with many important details that are missing nowadays.

Acknowledgement The authors acknowledge financial support from Russian Foundation for Basic Researches under Grant No. 05-0332719.

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Further reading D. Penn, Phys. Rev. 128 (1962) 2093.